Ordinary lattice defects as probes of topology

TL;DR

This work addresses how ubiquitous ordinary lattice defects can serve as universal probes of Bloch-band topology. By analyzing a square-lattice Qi-Wu-Zhang model and introducing vacancies, Schottky defects, substitutions, interstitials, and Frenkel pairs, the authors identify defect-bound mid-gap states whose existence encodes local topological information, with precise predictions supported by acoustic-lattice experiments using Green's-function spectroscopy. A key finding is that vacancies and Schottky defects reveal topology via internal boundaries only in topological phases, while substitutions, interstitials, and Frenkel pairs bind mid-gap modes regardless of global topology, and these defect states are robust to weak disorder. The results, demonstrated across theory and experiment, position ordinary defects as powerful, general tools for diagnosing topology and hint at defect-engineered platforms for localized Majorana modes and topological devices in various dimensions and symmetry classes.

Abstract

In addition to topological lattice defects such as dislocations and disclinations, crystals are also accompanied by unavoidable ordinary defects, devoid of any non-trivial geometry or topology, among which vacancies, Schottky defects, substitutions, interstitials, and Frenkel pairs are the most common. In this work, we demonstrate that these ubiquitous ordinary lattice defects, though topologically trivial, can nonetheless serve as universal probes of the non-trivial topology of electronic Bloch bands, and any change in the local topological environment in an otherwise normal insulator in terms of mid-gap bound states in their vicinity. We theoretically establish these generic findings by implementing a minimal model Hamiltonian describing time-reversal symmetry breaking topological and normal insulators on a square lattice, fostering such point defects. The defect-bound mid-gap modes are also shown to be robust against weak point-like charge impurities. Furthermore, we showcase experimental observation of such bound states by embedding ordinary crystal defects in two-dimensional acoustic Chern lattices, where precision-controlled hopping amplitudes are implemented via active meta-atoms and Green's-function-based spectroscopy is used to reconstruct spectra and eigenstates. Our combined theory-experiment study establishes ordinary lattice defects as probes of topology that should be germane in crystals of any symmetry and dimension, raising the possibility of arresting localized Majorana modes near such defects in the bulk of topological superconductors and to emulate ordinary-defect-engineered topological devices.

Figures (12)

• Figure 1: Schematic representations of (a) a vacancy, (b) a Schottky defect, (c) a substitution, (d) an interstitial, and (e) a Frenkel pair, where the defect cores are encircled by black dashed closed loops. The solid red and blue circles at each site of the underlying square lattice represent two orbitals with opposite parity eigenvalues ($+$ and $-$) of the Qi-Wu-Zhang model. To create a vacancy defect we eliminate both the orbitals from the same site, whereas to create a Schottky defect we eliminate complementary orbitals from two different sites of the square lattice. A substitution defect is generated by setting the value of the on-site orbital staggered potential $m_0$ at a given site in a topologically different parameter regime of the Qi-Wu-Zhang model (represented by the dashed red and blue circles) than the background one (solid red and blue circles). An interstitial defect is built by placing a site at an irregular point within the square lattice with a different on-site orbital staggered potential value in comparison to the background one. A Frenkel pair is a superposition of a vacancy and an interstitial. With a Frenkel pair defect, we average the local density of states of the resulting in-gap bound states (when they exist) over eight positions (marked by one clear and seven translucent dashed red and blue circles) of the constituting interstitial for a fixed vacancy position in our numerical simulations (but not in acoustic lattice-based measurements).
• Figure 2: (a) A schematic illustration of the unidirectional hopping implementation from Cavity 1 to Cavity 2, representing two sites or orbitals. The middle inset shows the equivalent tight-binding model with a representative hopping term of $i s t / 2$. A speaker (pump) placed in Cavity 1 provides excitations, while two microphones (probes) record the spectral responses in both cavities. (b) Photograph of the two coupled cavities (left). The amplifier controls the hopping strength, and the controller (right) functions as an integrated phase shifter, as illustrated in (a). Inset in the left panel: interior of the cavity cap, which houses two speakers and one microphone. (c) Experimentally measured (solid lines) and numerically fitted (dashed lines) amplitude (top) and phase (bottom) responses of the cross-power spectral density between the acoustic signals in Cavity 1 and Cavity 2. The target hopping is $i s t / 2 = -0.8 i$, and the fitted value is $-0.82 i$. (d) A schematic of a $7 \times 5$ acoustic lattice with a single vacancy at the center. Red and blue spheres denote two orbitals at each lattice site, while connecting tubes indicate hopping amplitudes. For details see Sec. \ref{['sec:experiment']}.
• Figure 3: Energy eigenvalues ($E_n$) as a function of its index ($n$) in the presence of a single vacancy [(a)-(d)] and five vacancies [(e)-(h)] located at the center of a square lattice with linear dimension $L=25 a$ and periodic boundary conditions in both directions, where $a$ is the lattice spacing. Inset in each panel shows the joint local density of states for two particle-hole symmetric closest-to-zero-energy modes (shown in red). The corresponding value of the mass parameter $m_0$ in the Qi-Wu-Zhang model and the associated Bott index (BI) are quoted in the legend. Therefore, the results are shown for the normal insulator with band minima near the ${\rm M}$ point [(a) and (e)], the ${\rm M}$ phase [(b) and (f)], the $\Gamma$ phase [(c) and (g)], and the normal insulator with band minima near the $\Gamma$ point [(d) and (h)]. See Fig. \ref{['fig:1']}(a) for reference and Sec. \ref{['subsec:vacancytheory']} for a detailed discussion on the results.
• Figure 4: Energy eigenvalues ($E_n$) as a function of its index ($n$) in the presence of a single Schottky defect for which the vacancies of opposite-parity orbitals reside within a distance $\sqrt{2} a$ [(a)-(d)] and $5 \sqrt{2}a$ [(e)-(h)] about the center of a square lattice with linear dimension $L=24 a$ and periodic boundary conditions in both directions, where $a$ is the lattice spacing. Inset in each panel shows the joint local density of states for two particle-hole symmetric closest-to-zero-energy modes (shown in red). The corresponding value of the mass parameter $m_0$ in the Qi-Wu-Zhang model and the associated Bott index (BI) are quoted in the legend. Therefore, the results are shown for the normal insulator with band minima near the ${\rm M}$ point [(a) and (e)], the ${\rm M}$ phase [(b) and (f)], the $\Gamma$ phase [(c) and (g)], and the normal insulator with band minima near the $\Gamma$ point [(d) and (h)]. See Fig. \ref{['fig:1']}(b) for reference and Sec. \ref{['subsec:schottkytheory']} for a detailed discussion on the results.
• Figure 5: Energy eigenvalues ($E_n$) as a function of its index ($n$) in the presence of a single substitution [(i.a)-(i.l)] and five substitutions [(ii.a)-(ii.l)] located at the center of a square lattice with linear dimension $L=25 a$ and periodic boundary conditions in both directions, where $a$ is the lattice spacing. Inset in each panel shows the joint local density of states for two particle-hole symmetric closest-to-zero-energy modes (shown in red). The corresponding value of the background mass parameter $m^{\rm back}_0$ on the parent square lattice sites and substituted mass parameter $m^{\rm sub}_0$ on the substitution defect sites in the Qi-Wu-Zhang model, respectively represented by the solid and dashed circles (red and blue) in Fig. \ref{['fig:1']}(c), and the associated Bott index (BI) are quoted in the legend of each panel. See Sec. \ref{['subsec:substitutiontheory']} for a detailed discussion on the results.